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Abstract

Background

The evolutionary success of Wolbachia bacteria, infections of which are widespread in invertebrates, is largely attributed
to an ability to manipulate host reproduction without imposing substantial fitness
costs. Here, we describe a stage-structured model with deterministic immature lifestages
and a stochastic adult female lifestage. Simulations were conducted to better understand
Wolbachia invasions into uninfected host populations. The model includes conventional Wolbachia parameters (the level of cytoplasmic incompatibility, maternal inheritance, the relative
fecundity of infected females, and the initial Wolbachia infection frequency) and a new parameter termed relative larval viability (RLV), which is the survival of infected larvae relative to uninfected larvae.

Results

The results predict the RLV parameter to be the most important determinant for Wolbachia invasion and establishment. Specifically, the fitness of infected immature hosts must
be close to equal to that of uninfected hosts before population replacement can occur.
Furthermore, minute decreases in RLV inhibit the invasion of Wolbachia despite high levels of cytoplasmic incompatibility, maternal inheritance, and low
adult fitness costs.

Conclusions

The model described here takes a novel approach to understanding the spread of Wolbachia through a population with explicit dynamics. By combining a stochastic female adult
lifestage and deterministic immature/adult male lifestages, the model predicts that
even those Wolbachia infections that cause minor decreases in immature survival are unlikely to invade
and spread within the host population. The results are discussed in relation to recent
theoretical and empirical studies of natural population replacement events and proposed
applied research, which would use Wolbachia as a tool to manipulate insect populations.

Background

The success of obligate endosymbiotic organisms depends on their ability to invade,
establish and persist in their host. Wolbachia pipientis, a well-studied endosymbiont, is a species of maternally inherited bacteria in the
order Rickettsiales, and infections are estimated to occur in more than half of all
insect species [1]. Prior studies have demonstrated the ability of Wolbachia to manipulate the reproduction of its host [2,3]; several phenotypes have been described, including male-killing [4,5], feminization [6,7], parthenogenesis [8-10], and cytoplasmic incompatibility (CI) [11-13]. CI affects a broad range of insect taxa and causes a reduction in egg hatch when
Wolbachia-uninfected females and Wolbachia-infected males mate (Figure 1).

Prior models highlight three Wolbachia-specific parameters that affect the probability of Wolbachia invasion and establishment: the maternal inheritance rate, which is the proportion
of infected offspring produced by an infected female; the level of CI, which is the
proportion of embryos that fail to develop as a result of incompatible crosses [14]; and the fitness cost to females for carrying a Wolbachia infection, defined as a decrease in overall fecundity [15-20].

Previous studies predict that the successful invasion of Wolbachia into an uninfected host population requires low fecundity costs, high maternal inheritance
rates, and high levels of CI [21,22]. Wolbachia infections that impose a 10% relative fecundity cost to adult females experience reductions
in their invasion success [21]. Similarly, low maternal inheritance reduces the probability of Wolbachia invasion [22]. Higher initial Wolbachia infection frequencies are predicted to increase the probability of population replacement,
which can offset the above costs [14]. Models have also addressed population structure at the adult stage, impacts on adult
survival, stochastic effects, and overlapping generations [14,21-25].

The relative importance of Wolbachia effects on immature life stages has not been assessed theoretically. This is despite
multiple examples demonstrating an effect of Wolbachia on immature hosts. In the stored product pest Liposcelis tricolor (Psocoptera: Liposcelidae), Wolbachia infections can decrease development periods and increase survivorship in some immature
life stages [26]. Other studies demonstrate negative impacts of Wolbachia infections on larval survival and development time [27,28]. Recent studies have determined that when intraspecific competition is intense, Wolbachia-infected mosquito larvae experience reduced survival [29,30].

To better understand population replacement by CI-inducing Wolbachia, we have evaluated both Wolbachia infection dynamics and host population dynamics using a model that includes deterministic
immature and adult male lifestages and a stochastic adult female lifestage. Since
Wolbachia are transmitted maternally, the sex and infection status of hosts are explicit, and
adult females are tracked individually. The focus of this modeling approach was to
investigate changes in the probability of population replacement resulting from varying
the relative larval viability (RLV), expressed as relative survival of infected to uninfected larvae. The results are
presented in context with traditional parameters: the rate of CI, maternal inheritance
(MI), the relative fecundity of infected females (RF), and the initial Wolbachia infection frequency (IF), on the probability of population replacement.

Methods

The model simulates a panmictic population that is closed to immigrants and emigrants.
Consistent with previous studies, the model assumes mating is random and that Wolbachia infection has no effect on mating success. Females in the model mate once immediately
upon reaching maturity. Adult survival is density-independent, but larval survival
is density-dependent. The model presented here combines a stochastic adult female
stage with deterministic adult male and immature stages. By implementing a deterministic
immature stage, additional information regarding population dynamics is incorporated
without developing a completely stochastic model, which would be considerably more
computationally-intensive. The model incorporates overlapping generations [24] while tracking major life stages and considers females and males separately. Development
time and survival during immature stages are addressed explicitly by the model. The
model was designed assuming the host is a holometabolous insect, and the model was
parameterized based upon estimates of mosquitoes in the genus Aedes as a case study.

Brief Description of Equations

The following is a brief overview of all equations and parameters implemented in the
model presented here. Additional development details, initial parameter values, and
sensitivity analysis are provided in Additional File 1.

Additional file 1.Model parameters, detailed equation appendix and sensitivity analysis. Portable Document File (pdf) containing all parameters, initial parameter values,
and equations utilized by the model. Model development is discussed, and includes
references from which each equation was developed/parameterized. Also includes the
sensitivity analysis of all population dynamic parameters and discussion about the
robustness of model predictions

Larval development rate R (developmental stage units): j is the maximum development rate (developmental stage units), h is the asymptotic minimum development rate (developmental stage units), Δt is the time step (units of time), q is the density-dependent development coefficient (units of (mass)-1), B is the total larval biomass (units of mass) and s is the total number of developmental stages. Derived from Gavotte et al. [30] and comparable to previously published data [31,32].

(2)

Larval survival, SL: μ is the baseline mortality rate of mosquito larvae in the absence of competition
(units of (time)-1). α is the coefficient controlling density dependent mortality (units of (time)-1). B is the total larval biomass (dimensionless), β is the exponent controlling density dependent mortality (dimensionless), γ is the coefficient that decreases mortality as development stage increases (units
of (time)-1), d is the developmental stage index, ε is the exponent that decreases mortality as development stage increases (dimensionless),
and Δt which is the time step (units of time). Based on Dye [33] and similar to previously published studies [34-36].

(3)

Mosquito body mass, M (units of mass): mx (units of mass) is the theoretical maximum mass of a given mosquito at time T. mx is linked to c (dimensionless), which is the percent of mx that is attainable. k (dimensionless) is the growth coefficient; T0 (dimensionless) is the development time at which mass at pupation is mx/2 days, and T (dimensionless) is development time. d (dimensionless) represents the total number of development stages completed by the
larval cohort. Derived from previously published data [30].

(4)

Female survivorship, Fs: g is the per capita mortality rate of adult females (units of (time)-1) and A is the current age of the female (units of time). Taken from Trpis and Hausermann
[37].

(5)

Egg production, E: u is the egg production rate; Δt is the time step (units of time); v is the female mass coefficient (units of (mass)-1); Mf is the body mass of the ovipositing female (units of mass); w is the female mass intercept (units of mass), and z is the female mass exponent (dimensionless). Derived by combining two previously published
functions [38,39].

Immature Life Stages

To simulate variation in egg hatch, the model assumes that some eggs (proportion equal
to H3, Table S1 in Additional file 1) hatch on day three while the remaining eggs (1-H3) hatch on day four (Figure 2a) [40,41]. Eggs are separated into two cohorts based on infection status. Larvae are distributed
into four categories for each of the possible combinations of sex and infection status.

Figure 2.Immature population structure. a) Eggs develop through four discrete stages and each stage is one day. There are
two cohorts of eggs, Wolbachia uninfected and infected. During development, eggs move through each stage consecutively,
and the number of eggs advancing to the next stage reflects the product of the number
of eggs present and SE, daily egg survivorship (Table S1). All eggs hatch after four days except a proportion
of eggs hatch at day three (H3, Table S1). b) Larvae develop through s discrete stages, where s is an arbitrary number of developmental stages (s = 30). Larvae are divided into four categories: Wolbachia infected/uninfected and male/female. Larvae move R developmental stages in each time step, where R is the number of developmental stages a larval cohort will progress (Equation 1).
The number of larvae progressing from their current development stage, e.g. L2, to their next developmental stage, L2+R, is equal to the product of the number of larvae in a developmental stage and larval
survival (Equation 2). Larval survival and development are density dependent. If larvae
are Wolbachia infected, they are subject also to the parameter RLV (Table 1), which can reduce the number of surviving larvae. Larvae that reach the
last developmental stage become pupae. c) Pupae progress through two discrete development
stages and are tracked similar to eggs. Each pupal developmental stage is one day
and pupae are subject to SP, daily pupal survivorship (Table S1).

Larvae develop through discrete developmental stages, where the development rate is
affected by density dependence, and larval survival is subject to both stage-dependent
mortality and density-dependence (Figure 2b). The term "stage" is defined here as a measure of progress through larval development.
The number of these discrete developmental stages is chosen to allow for variation
in development time and is otherwise arbitrary (i.e., not linked to age or developmental
instar explicitly). The number of larval developmental stages, s, can be varied, but was set to s = 30 for this study. Larval development rate, R, is the number of developmental stages through which a cohort of larvae will pass
within 24 hours (Equation 1). The number of larvae surviving to the next day is the
product of the number of larvae in the preceding time period and the larval survival
rate (Equation 2). When the number of developmental stages within a day is not an
integer, the larval cohort is distributed into two adjacent developmental stages in
proportions that preserve the average development rate. The latter also introduces
variation into the development rates of larval cohorts (Figure 2b). Density-dependence is based on the total mass of larvae (Equation 3). Male and
female cohorts are considered separately to observe sex-specific patterns during development.
For example, female mosquitoes require longer development time to become adults relative
to males, and studies demonstrate that males and females respond to competition intensities
differently [30].

Uninfected larval cohorts progress through development subject to stage-dependent
mortality and density dependent effects only. Infected larval cohorts are subject
also to a reduction in viability associated with Wolbachia infection. The relative larval viability (RLV, Table 1) for infected larvae is a proportion that indicates the relative survival of infected
to uninfected larvae.

Table 1. Glossary of notation, including the initial values for each key parameter

Following the completion of larval development stages, individuals become non-feeding
pupae, which have a daily survival that is independent of population density (Sp, Table S1; Figure 2c). After completing pupal development, emerging male adults are tracked separately
as either infected or uninfected cohorts. Emerging female adults are tracked as individuals.

Adult Life Stages

Six variables are tracked over time and determine the state of individual females:
the blood meal state (time since last feeding), age (days since emerging), Wolbachia infection status (infected or uninfected), the Wolbachia infection status of her mate (determined randomly based on the proportion of infected
males in the population at the time she mates), size (body mass), and reproductive
state (the number of gonotrophic cycles completed).

The probability that a female obtains a blood meal is determined by the frequency
of potential blood meals per unit area, and each blood meal is associated with an
additional mortality risk, regardless of mosquito age (Table S1). In the panmictic
population simulated here, the availability of potential blood meals is assumed to
be constant, but the model will allow downstream population structuring and geographic
variation of bloodmeal availability.

Adult female daily survivorship Fs is age-dependent and probabilistic (Equation 4) [37]. A female that is Wolbachia uninfected and mated with an infected male will lay eggs, but a proportion of the
eggs will not hatch, depending on the level of CI (Table 1). Infected females produce viable offspring regardless of their mate's infection
status but are subject to a decrease in relative fecundity (RF, Table 1). The number of eggs laid by an individual female is determined by her mass (Equation
5), and larval development influences female body mass. Specifically, intense competition
delays development and reduces the mass of adult females.

Adult males, which are dead end hosts for Wolbachia, are not tracked individually but are tracked as infected and uninfected cohorts.
The male mortality rate is assumed to be age-independent and constant (SM, Table S1). The proportion of Wolbachia infected males in the population determines the probability of an incompatible mating
for uninfected females.

Simulations

The model was written in MATLAB 7 (The MathWorks Inc., Natick, MA). A single simulation
of the model produced population dynamics that are tracked over time (Figure 3). A series of simulations (n = 1000) were used to assess the impact of incremental
parameter changes on the probability of population replacement. The parameters emphasized
were cytoplasmic incompatibility (CI); maternal inheritance (MI); the relative fecundity of adult females (RF); the initial Wolbachia infection frequency, expressed as a proportion of the total number of adults (IF), and the relative larval viability (RLV). A population replacement event is defined as having occurred when the proportion
of infected adults stabilizes above or equal to the MI value. During each series of simulations, individual parameters were varied singly,
while the remaining parameter values were held constant as defined in Table 1. Each parameter was uniformly varied at one one-hundredth intervals from zero to
one. At each interval, 1000 simulations were conducted, and the number of successful
invasions was recorded to determine the probability of population replacement at that
specific parameter value. The uniform sensitivity analysis was implemented for direct
comparisons between all parameters across all intervals. Furthermore, previous analyses
have not established minimum values for the spread of Wolbachia. Additional simulations tested two-way interactions between each of the emphasized
parameters by varying two parameters simultaneously and evaluating the probability
of population replacement. In the aforementioned simulations, parameters were varied
uniformly. One parameter would be held constant while the other parameter varied as
described above. The first parameter would then be incremented and the process above
would be repeated. The probabilities resulting from two-way interactions were approximately
the product of the two parameters and are not discussed further.

Figure 3.Example of typical population dynamics produced by a simulation of the model. a) Populations begin with an uninfected cohort of eggs. The population is allowed
to persist and self-regulate for 800 days, at which time Wolbachia is introduced to the population as gravid, bloodfed females at the rate defined in
Table 1. The population is then allowed to self-regulate and persist until 1800 days
have elapsed. b) The proportion of the female population that is infected with Wolbachia over time (i.e., infection frequency), demonstrating a population replacement event.

Results

Figure 3 provides an example of the typical population dynamics resulting from model simulations
of a Wolbachia population replacement event. In the illustrated example, the population begins as
cohort of uninfected eggs and stabilizes after approximately 150 days, with variation
around a consistent population size and lifestage distribution (Figure 3a). In the example simulation, the introduction of Wolbachia occurs at day 800 by introducing blood-fed, gravid adult females at an initial Wolbachia infection frequency (IF) of 0.5 (Table 1). IF is the frequency of Wolbachia-infected females relative to the total number of adults such that an IF = 1 is synonymous with a 1:1 (infected to uninfected) ratio. Figure 3b illustrates the resulting variation in Wolbachia infection frequency in the host population versus time.

Due to the stochastic nature of the model, the number of individuals within each lifestage
fluctuates considerably over time (Figure 3a). To examine for temporal patterns in the fluctuations that might correspond to periodic
signals such as stage durations or generation time, we performed a spectral analysis
on the time series data for both total adult and larval populations via Fast Fourier
Transformation [42,43]. The analysis can identify temporal patterns that exist in what appear to be chaotic
time series. No pattern was detected by the spectral analysis. Since no period was
found, stochasticity appears to be the sole driver of population fluctuations.

Five parameters associated with Wolbachia infection were evaluated for their affect on the probability of population replacement.
The value of each parameter was varied at one one-hundredth increments, from zero
to one, while additional parameters were held constant as defined in Table 1. For each parameter value, the probability of population replacement was determined
by the number of successful replacement events occurring in 1000 simulations, for
a total of 101,000 simulations per parameter.

Maternal inheritance (MI), the relative fecundity of adult females (RF), and relative larval viability (RLV), exhibit strong threshold behavior with population replacement occurring only at
parameter values exceeding 0.7 (Figure 4). Specifically, realistic probabilities of population replacement (i.e., > 50% probability
of population replacement) require the magnitude of MI to be greater than 0.9. Similarly, RF must exceed 0.9 before realistic probabilities of population replacement are attained.
The probability of population replacement is most sensitive to RLV, which requires a value of greater than 0.95 before population replacement can occur.
Furthermore, realistic probabilities of population replacement only occur at high
RLV (≥0.99), despite high maternal inheritance and CI (i.e., all other parameters held
at values defined in Table 1).

Figure 4.The probability of population replacement for five Wolbachia specific parameters. CI is the level of cytoplasmic incompatibility, MI is the level of maternal inheritance, IF is the initial frequency of Wolbachia infection, RF is the relative fecundity of Wolbachia-infected adult females, and RLV is the relative larval viability. Each line was generated by calculating the probability
of a population replacement event at one one-hundredth increments for parameter values
between zero and one (n = 1000 simulations/increment). IF and CI show similar responses to parameter value increases. The probability of population
replacement increases, but then asymptotically approaches one. The response curves
for RF, MI, and RLV behave similarly, each parameter requiring values to be greater than approximately
0.7. The curves then quickly increase toward one. RLV is the most sensitive parameter requiring values approaching 0.95 before a population
replacement event can occur. Realistic probabilities of population replacement (i.e.,
population replacement occurs in greater than 50% of simulations) does not occur until
RLV is greater than or equal to 0.99.

A different functional relationship is observed with the level of incompatibility
(CI) and initial Wolbachia infection frequency (IF), each of which results in response curves that increase asymptotically (Figure 4). Assuming the parameters within Table 1, the model predicts that CI is not necessary for Wolbachia to spread (i.e., approximately 7% of simulations resulted in population replacement
when CI = 0). Realistic probabilities of population replacement occur when CI approaches 0.3. Despite perfect CI (i.e., no egg hatch in incompatible crosses), population
replacement did not occur in 10% of simulations (Figure 4). Additional simulations confirmed that a 90% probability of population replacement
is an absolute maximum given the conditions defined here (Table 1). However, as the magnitude of IF increases, the probability of population replacement rapidly approaches one, with
realistic probabilities of population replacement occurring when the frequency of
infected females approaches 20% (Figure 4).

The results obtained from the model here were compared to a previously published stochastic
model [22]. Table 2 compares the fixation probabilities calculated by the model presented here and those
from Jansen et al. [22] using the conditions defined in the prior report, which includes the introduction
of a single infected female into a population size of 100 and perfect CI. To allow
direct comparison, the relative larval viability in our model was set to one. 50,000
simulations were performed for each combination of parameter values used in the prior
publication. Both models predict the probability of population replacement decreases
when MI and RF values are less than one (Table 2). Generally, our model predicted lower probabilities of population replacement than
the previously published model. However, when either MI or RF was 80%, the model presented here reported higher probabilities (Table 2). Jansen et al. [22] predicted that Wolbachia infections with imperfect maternal inheritance and low adult fitness costs (MI = RF = 0.9) will still invade and establish in a population, but our model predicted no
population replacement events (Table 2). The predictions of our model were also compared to those of Jansen et al. [22] assuming larger initial frequencies of Wolbachia infected individuals (Figure 5). Both models predict an asymptotic increase in the probability of population replacement
with increasing magnitude of IF, but our model predicts lower probabilities of population replacement (Figure 5).

Table 2. The probability of population replacement for given parameter values

Figure 5.The probability of population replacement by Wolbachia given different initial infection frequencies. This figure assumes that the relative fecundity of infected females is 0.95, with
perfect CI and maternal inheritance. The dashed line indicates the probability of
population replacement as calculated by Jansen et al [22], and the solid line represents the predictions of this model. Our model predicts
lower probabilities at all initial Wolbachia infection frequencies, but generates a similar functional response.

Discussion

The model presented here examines the probabilities of Wolbachia invasion into an isolated uninfected population. The model is unique in its individual-based
representation of variation in key traits among adult females and in the resolution
of larval dynamics within the host population. The model presented here predicts,
as in previous modeling studies, that maternal inheritance (MI) and the relative fecundity of adult females (RF) are key parameters that determine the potential for population replacement. Specifically,
population replacement occurs only at high MI or RF. In contrast, population replacement can occur at low CI or low IF. The simulation of adult females as individuals demonstrates that MI requires higher parameter values than RF for successful population replacement. The new parameter, relative larval viability
(RLV), like MI and RF, requires high parameter values before population replacement can occur.

The relative larval viability between Wolbachia infected and uninfected individuals (RLV) is the most important determinant of population replacement, requiring the highest
parameter values for invasion. The model predicts that reductions in infected larval
survival can substantially reduce the probability of population replacement (Figure
4). While a majority of prior studies have examined for an effect in adults, recent
studies have determined that, at high levels of intraspecific competition, Wolbachia infected larvae experience reduced survival [29]. However, few theoretical studies have examined the impact of immature lifestages
on the invasion of Wolbachia. Here, we demonstrate that reductions in RLV will inhibit Wolbachia invasion into an uninfected host population.

Recent work has highlighted the prevalence of Wolbachia, and its ability to invade populations [1,20]. Studies have suggested that Wolbachia infection affects larval survival and development only when intraspecific competition
is high [29,44]. Given the predictions from our model, Wolbachia can only invade a population when RLV is very high. Therefore, the density of conspecifics in larval habitats is predicted
to have significant impacts on the probability of population replacement. Similarly,
the abundance and variety of larval habitats may have significant impact on the invasion
of Wolbachia. The distribution, utilization and variety of larval habitats is well known for some
insects, particularly mosquitoes [45-48]. Theoretical studies considering the effect of metapopulation structure and larval
rearing conditions may elucidate the mechanism by which Wolbachia can invade natural populations given low initial infection frequencies.

The level of CI in insects varies widely [44,49-51]. Our model shows that the intensity of CI has relatively little effect on the probability
of population replacement when the rate of CI exceeds 60%. Furthermore, when CI = 0, the model presented here predicts population replacement can occur at low probabilities
(Figure 4). Some Wolbachia infections do not cause CI, but are found at high frequencies in natural populations
[44,50,52]. Previous theoretical studies indicate that CI or a sex-ratio distorter is not required
for population replacement when endosymbionts can alter female traits [44,53]. However, results presented here suggest that non-CI inducing Wolbachia infections can establish and persist in a population without increasing or altering
host fitness, given high MI, RF, and RLV. Since the population considered by the model presented here is relatively small
(N ≈ 110 adults), genetic drift could perhaps influence the probability of population
replacement [54]. To investigate the importance of genetic drift, the population size in the model
was increased. In model simulations where the total adult population size is greater
than approximately 200, population replacement does not occur when there is no effect
of CI (i.e. CI = 0). However, when population size is increased, the general response patterns in
Figure 4 are not altered.

High maternal inheritance rates have been observed consistently in natural populations
[55-57]. Furthermore, theoretical studies predict the probability of population replacement
declines as maternal inheritance decreases [12,21,22]. Similar to previous studies, results presented here suggest that maternal inheritance
(MI) must be high for a Wolbachia infection to invade an uninfected population and persist. Specifically, MI must be higher than 90% to attain a realistic probability of population replacement.

The effect of Wolbachia infections on adult female fitness has been well documented empirically and theoretically
[11,15,16,22,24,58,59]. Here, as in previous theoretical studies, the model predicts that the relative fecundity
of adult females (RF) must be high to facilitate population replacement.

For all parameters, the probability of population replacement approached an absolute
maximum of 90% given the conditions defined in Table 1. Here, the initially examined IF value is relatively high (0.5), analogous to artificial introductions examined in
prior theoretical work [25]. Subsequently, lower IF values have been simulated (Figure 4), including the introduction of a single, infected female (Table 2). The model predicts that Wolbachia invasion can occur at the lowest IF values and demonstrates an increasing probability of invasion with the higher introduction
levels, with the probability of population replacement approaching 100%. Additional
simulations determined that when IF is held constant and the total adult population size is increased, the probability
of population replacement approaches one given the conditions defined in Table 1. This result suggests genetic drift can affect the probability of population replacement
in small populations and may facilitate or hinder the spread of Wolbachia from low initial frequencies [54].

The model presented here predicted lower population replacement probabilities than
those predicted by previous stochastic models (Table 2 and Figure 5) [22]. Rasgon and Scott [25] noted a similar behavior where implementing population age-structure and overlapping
generations increased deterministic thresholds. The inclusion of additional life stages
and stage-structure in this stochastic model may explain the reduced probabilities
of population replacement. However, the model presented here predicted marginally
higher probabilities of population replacement when either maternal inheritance or
the relative fecundity of infected females had a magnitude of 0.8. The increased probability
of population replacement predicted by the model presented here is likely a result
of the individual-based representation of the adult female life stage that includes
stochastic survival.

The model here addresses a single, panmictic, isolated population but could be expanded
to include metapopulation structure. If introduction events can be assumed to occur
randomly, then the surrounding subpopulations should generally tend to inhibit population
replacement, because migration between subpopulations would dilute the proportion
of infected individuals. However, as demonstrated here, genetic drift may influence
the invasion of Wolbachia in smaller subpopulations. The spatial spread of Wolbachia has been assessed analytically by others and defines the conditions needed for Wolbachia to spread through space [20,24].

The majority of models that address the invasion of Wolbachia into uninfected populations have examined populations without lifestage subdivisions,
suggesting that additional empirical studies focused on understanding larval dynamics
are needed [34]. Many of the parameters defined here may be difficult to determine in natural populations
[25], but our results demonstrate the importance of understanding the role of life history
parameters and their interactions, despite the difficulties. Furthermore, the sensitivity
analysis of the model presented here demonstrates that the magnitudes of particular
parameters strongly influence the potential for spread and establishment of Wolbachia; these (e.g., Wolbachia effects on immature fitness) should be the focus of future empirical and theoretical
studies. Future theoretical studies could further address parameter sensitivity by
hyper-cube sampling, but this would require information about the distribution of
parameters to investigated [60].

Conclusions

Wolbachia is currently being utilized as the basis for a gene drive strategy in open field releases
of Aedes aegypti [61,62]; however, the predictions of the model presented here suggest that minute reductions
in RLV can inhibit population replacement. Research needs to focus on understanding the effects
of novel Wolbachia infections on immature lifestages. Xi et al. [63] demonstrated that novel Wolbachia infections can establish in a new host species and replace an uninfected population,
but the initial frequency of Wolbachia infected individuals needed to replace the population was higher than predicted. The
authors suggested that differences in survival of immature lifestages could explain
their results. Results presented here indicate that even reductions in RLV that are difficult to detect empirically will substantially reduce the probability
of population replacement.

The rapid decline in the probability of population replacement associated with reduced
larval viability indicates that empirical studies directed toward quantifying the
effects of endosymbionts on immature insects are important for understanding and predicting
Wolbachia invasion events. Recent empirical studies also suggest that a more complete understanding
of the effects of Wolbachia on the immature life stages is generally needed through additional empirical and theoretical
studies [28-30].

Authors' contributions

The original model was conceptualized by PRC, JWM and ES in a class taught by PHC.
The model was revised by YH, PHC, and SLD. JWM and ES parameterized equations, which
were derived by PHC. PRC coded the model and simulations in MATLAB, which was later
optimized by PRC, PHC, and YH. The manuscript was written by PRC, PHC, and SLD. All
authors have read and approved the final manuscript.

Acknowledgements

The authors would like to thank Michael Turelli and Peter Hammerstein for comments
and suggestions on this project. This research was supported by grants from the National
Institutes of Health [AI-067434] and the Bill and Melinda Gates Foundation [#44190].
This is publication 11-08-042 of the University of Kentucky Agricultural Experiment
Station.

References

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